Question

1. identify the segment bisector of \\(\overline{ab}\\). then find \\(am\\).

image: line segment \\(ab\\) with point \\(m\\), line \\(n\\) intersecting at \\(m\\), \\(am\\) labeled \\(2x + 10\\), \\(mb\\) labeled \\(5x - 23\\), and options:
a. line \\(n\\); \\(am = 11\\)
b. line \\(n\\); \\(am = 32\\)
c. \\(\overleftrightarrow{nm}\\); \\(am = 32\\)
d. \\(\overleftrightarrow{mn}\\); \\(am = 11\\)

1. the midpoint of \\(\overline{gh}\\) is \\(m(-1, -4)\\). one endpoint is \\(h(1, -3)\\). find the coordinates of endpoint \\(g\\).

Explanation:

Problem 7

Step1: Identify Bisector and Set Equation

Since line \( n \) bisects \( \overline{AB} \), \( AM = MB \). So \( 2x + 10 = 5x - 23 \).

Step2: Solve for \( x \)

Subtract \( 2x \) from both sides: \( 10 = 3x - 23 \).
Add 23 to both sides: \( 33 = 3x \).
Divide by 3: \( x = 11 \).

Step3: Find \( AM \)

Substitute \( x = 11 \) into \( AM = 2x + 10 \): \( AM = 2(11) + 10 = 22 + 10 = 32 \).
The bisector is line \( n \) (or \( \overleftrightarrow{NM} \) or \( \overleftrightarrow{MN} \); from options, \( \overleftrightarrow{NM} \) with \( AM = 32 \) matches option c).

Step1: Midpoint Formula

Midpoint \( M(x_m, y_m) \) of \( \overline{GH} \) with \( G(x_g, y_g) \) and \( H(x_h, y_h) \) is \( x_m = \frac{x_g + x_h}{2} \), \( y_m = \frac{y_g + y_h}{2} \).
Given \( M(-1, -4) \), \( H(1, -3) \), let \( G = (x, y) \).

Step2: Solve for \( x \)

\( -1 = \frac{x + 1}{2} \). Multiply by 2: \( -2 = x + 1 \). Subtract 1: \( x = -3 \).

Step3: Solve for \( y \)

\( -4 = \frac{y + (-3)}{2} \). Multiply by 2: \( -8 = y - 3 \). Add 3: \( y = -5 \).

Answer:

c. \( \overleftrightarrow{NM} \); \( AM = 32 \)

Problem 8